{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:OYJJVJ67RJWRJ6IY2JSEYV2YCC","short_pith_number":"pith:OYJJVJ67","canonical_record":{"source":{"id":"1710.07432","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-10-20T07:09:13Z","cross_cats_sorted":[],"title_canon_sha256":"3697941c2efdcc6572c890f378c39f50ac4170f3538696bf9afcf1c231439693","abstract_canon_sha256":"1c55cfb770e68af5fad97ba5d4f152ffa3715c412c33433e3481d72959865450"},"schema_version":"1.0"},"canonical_sha256":"76129aa7df8a6d14f918d2644c5758109b2f9c09c840db4f62a3f344006fcd90","source":{"kind":"arxiv","id":"1710.07432","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1710.07432","created_at":"2026-05-18T00:22:05Z"},{"alias_kind":"arxiv_version","alias_value":"1710.07432v2","created_at":"2026-05-18T00:22:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.07432","created_at":"2026-05-18T00:22:05Z"},{"alias_kind":"pith_short_12","alias_value":"OYJJVJ67RJWR","created_at":"2026-05-18T12:31:34Z"},{"alias_kind":"pith_short_16","alias_value":"OYJJVJ67RJWRJ6IY","created_at":"2026-05-18T12:31:34Z"},{"alias_kind":"pith_short_8","alias_value":"OYJJVJ67","created_at":"2026-05-18T12:31:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:OYJJVJ67RJWRJ6IY2JSEYV2YCC","target":"record","payload":{"canonical_record":{"source":{"id":"1710.07432","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-10-20T07:09:13Z","cross_cats_sorted":[],"title_canon_sha256":"3697941c2efdcc6572c890f378c39f50ac4170f3538696bf9afcf1c231439693","abstract_canon_sha256":"1c55cfb770e68af5fad97ba5d4f152ffa3715c412c33433e3481d72959865450"},"schema_version":"1.0"},"canonical_sha256":"76129aa7df8a6d14f918d2644c5758109b2f9c09c840db4f62a3f344006fcd90","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:22:05.520130Z","signature_b64":"BFqVvBowsgG6dhb3FxwH0Nnck2R5wDvEwAbtbVCbSquxa/StGyxU6oMY1inD0G9c5Qa1chSIHzIkpuO2YH8RDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"76129aa7df8a6d14f918d2644c5758109b2f9c09c840db4f62a3f344006fcd90","last_reissued_at":"2026-05-18T00:22:05.519651Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:22:05.519651Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1710.07432","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:22:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mxMC1I95/bXU6oTr9IPBYMluFzouLdsnluPpxwhjJKH3bhQ4PYvmmCvk4tX1FEHNEGc3ydAlwUb97+Z/wDZMCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T19:32:49.416072Z"},"content_sha256":"074c5ffc2f76ee569db39d91b3311634b19dda73ea277651c9bc52633069cbd8","schema_version":"1.0","event_id":"sha256:074c5ffc2f76ee569db39d91b3311634b19dda73ea277651c9bc52633069cbd8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:OYJJVJ67RJWRJ6IY2JSEYV2YCC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Extremal problems on saturation for the family of $k$-edge-connected graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Douglas B. West, Hui Lei, Suil O, Xuding Zhu, Yongtang Shi","submitted_at":"2017-10-20T07:09:13Z","abstract_excerpt":"Let $\\mathcal{F}$ be a family of graphs. A graph $G$ is $\\mathcal{F}$-saturated if $G$ contains no member of $\\mathcal{F}$ as a subgraph but $G+e$ contains some member of $\\mathcal{F}$ whenever $e\\in E(\\overline{G})$. The saturation number and extremal number of $\\mathcal{F}$, denoted $sat(n,\\mathcal{F})$ and $ex(n,\\mathcal{F})$ respectively, are the minimum and maximum numbers of edges among $n$-vertex $\\mathcal{F}$-saturated graphs. For $k\\in\\mathbb{N}$, let $\\mathcal{F}_k$ and $\\mathcal{F}'_k$ be the families of $k$-connected and $k$-edge-connected graphs, respectively. Wenger proved $sat(n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.07432","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:22:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9nWrDfb40nBlmtv2m5Zp8Unm5AHgX2W9Iaj4wSqllXe2gmIFdexSGuvldi91KSOUYo62OzQBFDfV3KThIVoGDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T19:32:49.416435Z"},"content_sha256":"c51e1e96299ecae377bd726e8fa9b7bfb1269360b55a8d0408fcb06659356847","schema_version":"1.0","event_id":"sha256:c51e1e96299ecae377bd726e8fa9b7bfb1269360b55a8d0408fcb06659356847"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OYJJVJ67RJWRJ6IY2JSEYV2YCC/bundle.json","state_url":"https://pith.science/pith/OYJJVJ67RJWRJ6IY2JSEYV2YCC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OYJJVJ67RJWRJ6IY2JSEYV2YCC/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-03T19:32:49Z","links":{"resolver":"https://pith.science/pith/OYJJVJ67RJWRJ6IY2JSEYV2YCC","bundle":"https://pith.science/pith/OYJJVJ67RJWRJ6IY2JSEYV2YCC/bundle.json","state":"https://pith.science/pith/OYJJVJ67RJWRJ6IY2JSEYV2YCC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OYJJVJ67RJWRJ6IY2JSEYV2YCC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:OYJJVJ67RJWRJ6IY2JSEYV2YCC","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"1c55cfb770e68af5fad97ba5d4f152ffa3715c412c33433e3481d72959865450","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-10-20T07:09:13Z","title_canon_sha256":"3697941c2efdcc6572c890f378c39f50ac4170f3538696bf9afcf1c231439693"},"schema_version":"1.0","source":{"id":"1710.07432","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1710.07432","created_at":"2026-05-18T00:22:05Z"},{"alias_kind":"arxiv_version","alias_value":"1710.07432v2","created_at":"2026-05-18T00:22:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.07432","created_at":"2026-05-18T00:22:05Z"},{"alias_kind":"pith_short_12","alias_value":"OYJJVJ67RJWR","created_at":"2026-05-18T12:31:34Z"},{"alias_kind":"pith_short_16","alias_value":"OYJJVJ67RJWRJ6IY","created_at":"2026-05-18T12:31:34Z"},{"alias_kind":"pith_short_8","alias_value":"OYJJVJ67","created_at":"2026-05-18T12:31:34Z"}],"graph_snapshots":[{"event_id":"sha256:c51e1e96299ecae377bd726e8fa9b7bfb1269360b55a8d0408fcb06659356847","target":"graph","created_at":"2026-05-18T00:22:05Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $\\mathcal{F}$ be a family of graphs. A graph $G$ is $\\mathcal{F}$-saturated if $G$ contains no member of $\\mathcal{F}$ as a subgraph but $G+e$ contains some member of $\\mathcal{F}$ whenever $e\\in E(\\overline{G})$. The saturation number and extremal number of $\\mathcal{F}$, denoted $sat(n,\\mathcal{F})$ and $ex(n,\\mathcal{F})$ respectively, are the minimum and maximum numbers of edges among $n$-vertex $\\mathcal{F}$-saturated graphs. For $k\\in\\mathbb{N}$, let $\\mathcal{F}_k$ and $\\mathcal{F}'_k$ be the families of $k$-connected and $k$-edge-connected graphs, respectively. Wenger proved $sat(n","authors_text":"Douglas B. West, Hui Lei, Suil O, Xuding Zhu, Yongtang Shi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-10-20T07:09:13Z","title":"Extremal problems on saturation for the family of $k$-edge-connected graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.07432","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:074c5ffc2f76ee569db39d91b3311634b19dda73ea277651c9bc52633069cbd8","target":"record","created_at":"2026-05-18T00:22:05Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"1c55cfb770e68af5fad97ba5d4f152ffa3715c412c33433e3481d72959865450","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-10-20T07:09:13Z","title_canon_sha256":"3697941c2efdcc6572c890f378c39f50ac4170f3538696bf9afcf1c231439693"},"schema_version":"1.0","source":{"id":"1710.07432","kind":"arxiv","version":2}},"canonical_sha256":"76129aa7df8a6d14f918d2644c5758109b2f9c09c840db4f62a3f344006fcd90","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"76129aa7df8a6d14f918d2644c5758109b2f9c09c840db4f62a3f344006fcd90","first_computed_at":"2026-05-18T00:22:05.519651Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:22:05.519651Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BFqVvBowsgG6dhb3FxwH0Nnck2R5wDvEwAbtbVCbSquxa/StGyxU6oMY1inD0G9c5Qa1chSIHzIkpuO2YH8RDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:22:05.520130Z","signed_message":"canonical_sha256_bytes"},"source_id":"1710.07432","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:074c5ffc2f76ee569db39d91b3311634b19dda73ea277651c9bc52633069cbd8","sha256:c51e1e96299ecae377bd726e8fa9b7bfb1269360b55a8d0408fcb06659356847"],"state_sha256":"bef51916b9904837a3a9b8919382f0d602c444b45b72ec81ce41822db558695d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/e1utT3I04IJyQCYmQb3cQcLndnzdcBH+Bi9ndpW95EXHhSNXJlH7ztojE56UMjavzfn/SdIhStYWxeIUUaACw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T19:32:49.418447Z","bundle_sha256":"c05efac7531e1ba0c55d87afe58e004308ea2bc5d2d498f05b26221c520f9ff3"}}